| L(s) = 1 | + (−1.26 + 1.55i)2-s + (−0.806 − 3.91i)4-s + (−1.35 − 2.34i)5-s + (10.0 + 5.79i)7-s + (7.09 + 3.70i)8-s + (5.35 + 0.865i)10-s + (8.54 + 4.93i)11-s + (0.296 + 0.513i)13-s + (−21.6 + 8.24i)14-s + (−14.6 + 6.31i)16-s + 8.87·17-s + 14.0i·19-s + (−8.10 + 7.20i)20-s + (−18.4 + 7.01i)22-s + (18.2 − 10.5i)23-s + ⋯ |
| L(s) = 1 | + (−0.631 + 0.775i)2-s + (−0.201 − 0.979i)4-s + (−0.271 − 0.469i)5-s + (1.43 + 0.828i)7-s + (0.886 + 0.462i)8-s + (0.535 + 0.0865i)10-s + (0.777 + 0.448i)11-s + (0.0227 + 0.0394i)13-s + (−1.54 + 0.588i)14-s + (−0.918 + 0.394i)16-s + 0.522·17-s + 0.742i·19-s + (−0.405 + 0.360i)20-s + (−0.838 + 0.318i)22-s + (0.794 − 0.458i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.987590 + 0.485516i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.987590 + 0.485516i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 - 1.55i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.35 + 2.34i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-10.0 - 5.79i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.54 - 4.93i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-0.296 - 0.513i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 8.87T + 289T^{2} \) |
| 19 | \( 1 - 14.0iT - 361T^{2} \) |
| 23 | \( 1 + (-18.2 + 10.5i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (10.1 - 17.6i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-14.3 + 8.27i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 40.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + (21.2 + 36.7i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (32.2 + 18.6i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (1.57 + 0.907i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 21.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (76.6 - 44.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-36.4 + 63.2i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (38.3 - 22.1i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 111. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 76.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + (8.30 + 4.79i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (73.6 + 42.5i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 64.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + (3.59 - 6.22i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.10475057860647059321427218474, −12.39831155175270163498284524754, −11.50539383037959967871806778910, −10.26358864100356766504868716360, −8.880229872104417116596001894549, −8.338815806177084170122802843446, −7.11173488768734900408786342500, −5.59482071114531624799244949933, −4.58265542036218223704861650977, −1.58243423292837904645637517620,
1.32359757136277887153429086983, 3.37673861890857071640482653232, 4.74210282686623827295023809942, 7.00963441947485280818103989243, 7.932580486173425985253322414006, 9.013414808366224390727828534266, 10.37082164571775500307908742920, 11.24578278578339375537324254377, 11.75186935786872588133994465577, 13.29506245696106929012096953107
